I agree that a real time clock module is the way to go. These keep accurate time (as accurate as a typical watch/clock) and they have a backup battery so they don't have to be reset if you loose power. You can find examples of reading the clock with the Arduino.
The Arduino by itself can be used to take some action every 12 hours, but the time will tend to drift more than a RTC module. And unless you add a battery backup, the time will reset if you loose power.
Dimming an LED is straightforward and there are examples, but you'll have to modify the example for your desired 10-minute dim period.
You'll also need to decide how you are going to program the time and dim-actions. The hardware & software is easier if you use a computer. If you want to make a stand-alone system (like an alarm clock), you'll need an LCD display, some buttons, and all of the supporting software.
Would it be possible to use surface-mount LEDs?
Or, possibly, to apply the same concept to a panel of through-hole LEDs?
You said you were going to use LED strips... If you are using LED strips, they will usually have surface mount LEDs inside, but that's not important...
Different LEDs & different LED strips have different electrical requirements -
Standard individual LEDs (surface mount or through-hole) are normally driven directly through a current-limiting resistor. The Arduino can only power about one LED per I/O pin, so with several LEDs around the ceiling you'll need a transistor or MOSFET.
With transistors/MOSFETs you can power multiple LEDs (each with a resistor) in parallel, or with higher voltage in a series-parallel configuration.
LED strip sometimes just have series resistors. Other LED strips have transistors/MOSFETs built-in. Some have daisy-chained drivers so that all of the LEDs are individually addressable.
If it only has resistors, you need a power supply and a transistor/MOSFET (or 3 transistors/MOSFETs for RGB strips).
If the strip has active electronics built-in, you just supply power and you can control it directly from an Arduino I/O pin.
High power LEDs (1W or more) are generally run from a "constant current" power supply which you can build or buy. Commercial dimmable constant-current LED power supplies use a 0-10 VDC control-voltage (or 10V PWM) for dimming. With the Arduino, you need a 10V power supply and a small (low power) transistor or MOSFET to boost the Arduino's 5V PWM.
LED "light bulbs" from a hardware store usually run from 120/240 VAC or 12VAC. Some are dimmable, and some are not. They have active electronics built-in, and even the dimmable ones are not easily dimmable with the Arduino. They are normally dimmed with a phase-controlled dimmer (a normal AC light dimmer).
Many years ago, I built a "sunrise dimmer" with a different microcontroller. I'm still using it. The AC power is turned on & off with an [u]X-10[/u] relay and a programmable X-10 controller (part of my home automation system). The part I built doesn't have a real-time clock. It simply plugs into the X-10 controlled outlet and it starts dimming-up a regular AC powered incandescent light as soon as power is applied. (And mine doesn't dim-down... It stays on for about an hour 'till the master X-10 controller switches off power.)